Lie-Rinehart bialgebras for crossed products

Let g be a Lie algebra over a field K. Endow the polynomial ring K[t] with a g-action by derivations and consider the resulting crossed product K[t]⊙θg in the category of (K,K[t])-Lie algebras. We explore Lie-Rinehart bialgebra structures on (K[t],K[t]⊙θg) that arise via compatible pairs and ε-dynamical r-matrices. We give a complete classification for .